Fix definition of 'less than' to not involve a third variable.

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice.agda

@@ -19,7 +19,7 @@ record IsSemilattice {a} (A : Set a)
     (_⊔_ : A → A → A) : Set a where
 
     _≼_ : A → A → Set a
-    a ≼ b = Σ A (λ c → (a ⊔ c) ≈ b)
+    a ≼ b = (a ⊔ b) ≈ b
 
     _≺_ : A → A → Set a
     a ≺ b = (a ≼ b) × (¬ a ≈ b)
@@ -34,11 +34,22 @@ record IsSemilattice {a} (A : Set a)
 
     open IsEquivalence ≈-equiv public
 
+    open import Relation.Binary.Reasoning.Base.Single _≈_ ≈-refl ≈-trans
+
     ≼-refl : ∀ (a : A) → a ≼ a
-    ≼-refl a = (a , ⊔-idemp a)
+    ≼-refl a = ⊔-idemp a
 
     ≼-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
-    ≼-cong a₁≈a₂ a₃≈a₄ (c₁ , a₁⊔c₁≈a₃) = (c₁ , ≈-trans (≈-⊔-cong (≈-sym a₁≈a₂) ≈-refl) (≈-trans a₁⊔c₁≈a₃ a₃≈a₄))
+    ≼-cong {a₁} {a₂} {a₃} {a₄} a₁≈a₂ a₃≈a₄ a₁⊔a₃≈a₃ =
+        begin
+            a₂ ⊔ a₄
+        ∼⟨ ≈-⊔-cong (≈-sym a₁≈a₂) (≈-sym a₃≈a₄) ⟩
+            a₁ ⊔ a₃
+        ∼⟨ a₁⊔a₃≈a₃ ⟩
+            a₃
+        ∼⟨ a₃≈a₄ ⟩
+            a₄
+        ∎
 
     ≺-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≺ a₃ → a₂ ≺ a₄
     ≺-cong a₁≈a₂ a₃≈a₄ (a₁≼a₃ , a₁̷≈a₃) =